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I have a varchar field, which is also a lookup field.

I am enabling Extended search on this field, but Search Panel is not displaying.
No Console error. PHP Maker Version 2024.4

Did you also enable "Filter" for this field? From the docs:
Note This setting is mutually exclusive with Ext. Search. If both are enabled, Filter will be used.

Both are enabled.
Filter option is working fine but extended search is not displaying.

As mentioned by documentation above:
Note This setting is mutually exclusive with Ext. Search. If both are enabled, Filter will be used.

When you enable both Ext. Search and Filter, then only Filter will be used. The Ext. Search for that field won't be displayed.